In a letter to the Homeland Security secretary, Mamdani says the city has not authorized ICE use and will go to court if it continues.
A judge sets bond at $30,000 for Mario Alberto Morales, whose lawyers call the case political, while the Harris County district attorney says it is not.
DHS says it is working to address a potential breach, and a group calling itself the Department of Peace claims it took the data.
More than a dozen news organizations want the warrant released, but prosecutors say disclosure could jeopardize the ongoing criminal investigation of the Minnesota Democrat.
Axios reports the push also covers cutting aid to Israel, and says moderate Democrats have long opposed defunding ICE.
The Daily Wire reports, citing a court letter it obtained, that the inquiry follows a complaint over her conviction for obstructing ICE officers at a courthouse.
Staten Island's district attorney's office calls the report false and political, while Manhattan's office says its 2022 policy on immigration consequences is outdated.
Harris County's district attorney says the officer steered at the man on purpose, but ICE denies that and calls the case "a political stunt."